Tactics
The x-ray
A piece attacks or defends through another piece. It looks like nothing while the line is blocked and becomes real the moment the piece between them moves or is taken, which means an x-ray is a threat that already exists and has not been noticed yet.
How to spot it
When you count attackers and defenders on a square, count the pieces behind the ones you can see. A queen behind a rook on the same file is two attackers, not one, and an enemy rook that appears to be defended is not defended if the defender is standing on the line you are attacking along.
The positions
Easiest first. Every one is from a real game, and the rating under each board is what it costs the players who have tried it on lichess.
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1. Black to play Play it out →Answer
24...Qxe1+ 25.Rxe1 Rxe1+ 26.Qf1 Rxf1#
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2. White to play Play it out →Answer
35.Qd8+ Rxd8 36.Rxd8#
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3. Black to play Play it out →Answer
30...Ra8 31.Qxa2 Qxa2 32.Rxa2 Rxa2
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4. Black to play Play it out →Answer
36...Qxe2 37.Rxe2 Rxe2
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5. Black to play Play it out →Answer
34...Qb1+ 35.Bc1 Qxc1+ 36.Rxc1 Rxc1+ 37.Qd1 Rxd1+ 38.Re1 Rxe1#
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6. Black to play Play it out →Answer
40...Qxf2+ 41.Kh1 Qf3+ 42.Bxf3 Bxf3#
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7. Black to play Play it out →Answer
28...Qf3+ 29.Bxf3 Bxf3+ 30.Kg1 Rg8+ 31.Ng4 Rxg4+ 32.Kf2 Rg2+ 33.Kxf3 Rxc2
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8. Black to play Play it out →Answer
16...Qxf2+ 17.Kh1 Qf3+ 18.Bxf3 Bxf3#
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9. Black to play Play it out →Answer
29...Qd6 30.Re8+ Bg8 31.Rxg8+ Rxg8
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10. White to play Play it out →Answer
24.Qa8+ Qb8 25.Rc8+ Re8 26.Rxe8+ Qxe8 27.Qxe8#
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11. Black to play Play it out →Answer
22...Ne2+ 23.Kh1 Rxh2+ 24.Qxh2 Qxh2+ 25.Kxh2 Rh8#
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12. White to play Play it out →Answer
15.Qa8+ Kd7 16.Bb5+ c6 17.Bxc6+ bxc6 18.Qxc6#
Reading about a pattern is the easy half. Motify finds the positions where you missed one, by running your own lichess and Chess.com games through Stockfish, and puts each of them behind two or three puzzles on the same idea so you meet it having just found it three times. See how Motify works.
Positions from the lichess puzzle database, released under CC0.